Positive integer solutions of the diophantine equation x 2 − L n xy + (−1) n y 2 = ±5 r

In this paper, we consider the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r and determine the values of n for which the equation has positive integer solutions x and y. Moreover, we give all positive integer solutions of the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r when the equation has positive integer solutions.     

Revealing the Spin-Hamiltonian and Characteristic Parameters of Paramagnetic Centers in Cu2+-Doped Potassium Metabisulfite (K2S2O5) Single Crystal by Electron Paramagnetic Resonance Technique

The present study is mainly related to finding out spin-Hamiltonian parameters of Cu²⁺-doped potassium metabisulfite [K₂S₂O₅] single crystals using electron paramagnetic resonance (EPR) technique which has been applied in the temperature range from 297 to 113 K for single and powder crystals of the title compound. The existence of two complexes and two Cu²⁺ sites for each complex was concluded from the angular variation of the EPR spectra. Having constructed the g and the hyperfine tensors, the spin-Hamiltonian parameters have been obtained for these two complexes. Using these parameters, the covalency parameter (α ²), mixing coefficients (α and β) and Fermi contact term (K) have been calculated.     

DIBUTADIENYL PIPERAZINE,DIBUTADIENYL HOMOPIPERAZINES AND BUTADIENYL-SUBSTITUTED PIPERIDINES FROM MONOSUBSTITUTED HALODIENES

Compounds 3a–k were obtained from the reactions of compounds 1a–k with homopiperazine (2) in CH ₂ Cl ₂ . Compounds 1a–b, 1d–f, and 1h–l gave compounds 5a–b, 5d–f, and 5h–l with 2-methylpiperazine (4) in dichloromethane. Compounds 7c and 9c were obtained from the reactions of compound 1c with 4-ethoxycarbonyl piperazine (6) and 4-piperidinol (8) in CH ₂ Cl ₂ . Compounds 1a and 1f gave compounds 11a and 11f with 4-methylpiperazine (10), and compound 13f was obtained from the reactions of compound 1f with 4-methylpiperidine (12) in CH ₂ Cl ₂ .     

New N,S-Substituted Dienes from the Reactions of Some Aliphatic Mono(thio)substituted Nitrodienes with Aromatic Primary Amines and Cyclic Amines

Reaction of 2-nitropentachlorobutadiene with thiols and amines gave new N,S- and S,S-substituted nitrodiene compounds is discussed.     

Synthesis,Characterization, and Spectroscopic Properties of Hexa(4-Bromo-2-Formyl-Phenoxy)Cyclotriphosphazene and Hexa(4-Chloro-2-Formyl-Phenoxy)Cyclotriphosphazene and Fully Substituted Cyclotriphosphazene Derivatives Bearing a Schiff Base at Room Temperature

Abstract

Hexa(4-bromo-2-formyl-phenoxy)cyclotriphosphazene (2) and hexa(4-chloro-2- formyl-phenoxy)cyclotriphosphazene (3) were obtained from the reactions of hexachloro- cyclotriphosphazene (1) with 5-bromosalicylaldehyde and 5-chlorosalicylaldehyde in the presence of (C₂H₅)₃N and K₂CO₃ at room temperature, respectively. The new two organocyclotriphosphazenes bearing formyl groups were reacted with 4-cyano aniline, 2-phenyl aniline, 4-aceto aniline, 5-chloro-2-hydroxy aniline, 2-hydroxy aniline, 4-hydroxy aniline, 2-(4-morpholino)ethyl amine, 4-carboxy aniline, 4-carbomoyl aniline, 2-mercapto aniline, and 5-amino isoquonoline to prepare cyclotriphosphazene derivatives containing a Schiff base at room temperature. However, fully phenoxy-substituted cyclotriphosphazenes containing a Schiff base were isolated from the reactions of the compound 2 and 3 with 5-chloro-2-hydroxy aniline, 2-hydroxy aniline, 4-hydroxy aniline, and 2-(4-morpholino)ethyl amine. The structures of the synthesized compounds were characterized by elemental analysis, IR, and NMR (¹H, ¹³C, ³¹P) spectroscopy. According to the results of the analysis, all synthesized compounds were found to be fully substituted organocyclotriphosphazenes, such as hexa[4-bromo-2-(5-chloro-2-hydroxy-pheyliminomethyl)phenoxy]cyclotriphosphaze (2a). All cyclotriphosphazene derivatives synthesized gave fluorescence emission peaks in range between 300 nm and 410 nm.